Holographic Equidistribution

TL;DR

The paper investigates how large-N Hecke operators acting on 2d CFT partition functions enact a universal modular averaging that eliminates heavy-state contributions, leaving a light-state Poincaré series with a natural holographic interpretation as a sum over semiclassical handlebody geometries. By combining the splitting of partition functions into a modular completion of light states with the equidistribution theorem, the authors apply this to code CFT ensembles, cyclic product orbifolds, and symmetric product orbifolds, showing that the large-N limit reduces to Poincaré-series terms built from light data while heavy data average to constants. They further extend the analysis to stringy unitarity considerations in AdS3 gravity through a Hecke image of a primitive partition and discuss potential ergodic interpretations and bulk-ensemble connections. The results provide a coherent number-theoretic mechanism for holographic averaging and bulk geometry emergence, with implications for BTZ universality, higher-group Hecke actions, and connections to L-functions and ergodic theory. Overall, lim_{N→∞} T_N Z(τ) ≈ T_N Z_hat_L(τ) + const plus a vanishing remainder, tying heavy spectral data to light-state modular images and supporting a semiclassical bulk picture.

Abstract

Hecke operators acting on modular functions arise naturally in the context of 2d conformal field theory, but in seemingly disparate areas, including permutation orbifold theories, ensembles of code CFTs, and more recently in the context of the AdS$_3$/RMT$_2$ program. We use an equidistribution theorem for Hecke operators to show that in each of these large $N$ limits, an entire heavy sector of the partition function gets integrated out, leaving only contributions from Poincaré series of light states. This gives an immediate holographic interpretation as a sum over semiclassical handlebody geometries. We speculate on further physical interpretations for equidistribution, including a potential ergodicity statement.

Figures (2)

• Figure 1: Density plots in the complex $q=e^{2\pi i \tau}$ Poincaré disk comparing the (holomorphic) torus partition function $Z_{\text{boson}}(\tau)$ of the chiral$c=1$ compact boson with the Hecke operator $T_{16}$ acting on $Z_{\text{boson}}(\tau)$. Note that every pole in the plot now has a greater winding number, in particular the $T\to\infty$ divergence at $q\to1$ has its winding number multiplied by 16, since the Hecke operator multiplies the effective central charge by $N=16$. We use these plots to illustrate the effect of a Hecke operator on a modular function, but we will work exclusively with nonchiral (nonholomorphic) partition functions otherwise.
• Figure 2: An analogy between density plots of the action of Hecke operators $T_N$ on the torus partition function $Z(\tau)$ of the chiral boson, and the action of the Baker's map $B(x,y)$ (a classical chaotic map) on a gray square gradient.